Algebraic curves, arcs, and caps over finite fields - 1986

J. W. P. Hirschfeld Full Version (PDF)
(Quaderni di Matematica, 5 / 1986)
QdM_5_1986 - Cover

These notes give an account of a series of lectures at the University of Lecce as well as two at the University of Bari, all during Apri1 1986.
§§1-15 are based on the thesis [18], of J.-F.Voloch, a part from some background remarks and classical interpolations. They deal with the number of points on an algebraic curve over a finite field. The main results of the thesis are also contained in [14], §16 records some classical results on elliptic curves and §17, following Voloch [19], proves the existence of complete k-arcs form any values of k by taking half the points on anelliptic curve. §§18-19 discusses the values of n(2,q), the size of the smallest k- art in PG(2,q), and m'(2,q), the size of the second largest complete k- art in PC(2,q), the main result of §19 follows a proof of Segre using an improved bound for the number of points on a curve from §§ll and 14. Finally, §20 summarizes the best, known estjmates for $m_2(d,q)$, the irrgest size of k- cap in PG(d,q).







Table Of Contents


Introduction     PDF

The maximum number of points on an algebraic curve     PDF
J. W. P. Hirschfeld 1-1

The deduction of Serre's and Ihara's results from the Riemann hypothesis     PDF
J. W. P. Hirschfeld 1-4

The essential idea in a particular case     PDF
J. W. P. Hirschfeld 4-7

Weierstrass points in characteristic zero     PDF
J. W. P. Hirschfeld 7-8

Fundamental definitions in algebraic geometry     PDF
J. W. P. Hirschfeld 8-11

The canonical series     PDF
J. W. P. Hirschfeld 10-11

The osculating hyperplane of a curve     PDF
J. W. P. Hirschfeld 11-14

The generalized Wronskian     PDF
J. W. P. Hirschfeld 15-17

Construction of some linear systems     PDF
J. W. P. Hirschfeld 18-19

The essential construction     PDF
J. W. P. Hirschfeld 19-24

Elliptic curves     PDF
J. W. P. Hirschfeld 24-25

Hyperelliptic curves     PDF
J. W. P. Hirschfeld 25-26

Plane curves     PDF
J. W. P. Hirschfeld 26-28

The maximum number of points on an algebraic curve     PDF
J. W. P. Hirschfeld 28-29

Elliptic curves: fundamental aspects     PDF
J. W. P. Hirschfeld 30-35

k-arcs on elliptic curves     PDF
J. W. P. Hirschfeld

k-arcs in pg (2,q)     PDF
J. W. P. Hirschfeld 42-43

An improvement on the bound for m^1(2,q) when q is prime     PDF
J. W. P. Hirschfeld 44-46

k-caps in PG (n,q), n\geq 3     PDF
J. W. P. Hirschfeld 46-47

Bibliography     PDF
J. W. P. Hirschfeld 48-49


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